Definition:Either-Or Topology

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Definition

Let $S = \left[{-1 \,.\,.\, 1}\right]$ be the closed interval on the real number line from $-1$ to $1$.


Let $\tau \subseteq \mathcal P \left({S}\right)$ be a subset of the power set of $S$ such that, for any $H \subseteq S$:

$H \in \tau \iff \left({\left\{{0}\right\} \nsubseteq H \lor \left({-1 \,.\,.\, 1}\right) \subseteq H}\right)$

where $\lor$ is the inclusive-or logical connective.


Then $\tau$ is the either-or topology, and $T = \left({S, \tau}\right)$ is the either-or space


That is, a set is open in $\tau$ if it does not contain $\left\{{0}\right\}$ or it does contain $\left({-1 \,.\,.\, 1}\right)$.


Also see

  • Results about the either-or topology can be found here.


Sources